A Dozen New Homework Problems

Russ Hobbie and I are hard at work on the 5th edition of Intermediate Physics for Medicine and Biology. Sometimes we consider adding new material, try things out, debate its merits, but in the end it doesn’t make the cut. For instance, we thought about adding a section on elasticity theory to Chapter 1, but that was going to be too long (we are constantly battling between adding important topics and keeping the book from getting too fat), so we tried writing some new homework problems to teach the material that way. But it was also too much, and eventually we gave up on the idea. For those wanting to learn more about the biological applications of elasticity theory, I recommend Y. C. Fung’s book Biomechanics: Mechanical Properties of Living Tissue (cited in IPMB), or his more general textbook First Course in Continuum Mechanics.

I don't want to see good homework problems go to waste, so I offer them here in this blog. Twelve new homework problems. Free. They offer a way to learn a bit of elasticity theory. Enjoy.

Problem 1 Consider the rod in Fig. 1.20; the x-axis is along the rod’s length and x=0 is where the rod meets the wall. Let the displacement u(x,y) be the change in position of each point in the material in response to the force F. Express the displacement as u_x=Ax, u_y=0, and u_z=0, where A is a constant.
(a) Calculate the normal strain ε_n using Eq. 1.24 and Fig. 1.20.
(b) Calculate ε_n using the definition ε_n=∂u_x/∂x . Is it the same as in (a)?

Problem 2 Consider the rod in Fig. 1.23; x is horizontal, y is vertical, and y=0 is where the rod meets the floor. Express the displacement as u_x=By, u_y=0, and u_z=0, where B is a constant.
(a) Calculate the shear strain ε_s using Eq. 1.27 and Fig. 1.23 (assume B « 1)
(b) Calculate ε_s using the definition ε_s=∂u_x/∂y+∂u_y/∂x. Is it the same as in (a)?

Problem 3 The normal and shear strains can be combined into a strain tensor, which a 3 x 3 matrix. Define the diagonal components of this tensor (ε_xx, ε_yy, ε_zz) as the normal strain in each direction, and the off-diagonal components (ε_xy, ε_yz, ε_zx) as one half of the shear strain in each direction. For example, ε_xy=(∂u_x/∂y+∂u_y/∂x)/2.
(a) Derive expressions relating each component of the strain tensor to the displacement.
(b) Show that the strain tensor is symmetric (e.g., ε_xy=ε_yx).
Note: This expression for the strain tensor is correct for small strains. For large strains it is a more complicated nonlinear function of the displacement (Fung, 1993).

Problem 4 The dilatation is defined as the change in volume over the original volume, ΔV/V. For small strains, the dilatation is ε_xx+ε_yy+ε_zz .
(a) Calculate the dilatation for the displacement in Problem 1. Does the volume change?
(b) Calculate the dilatation for the displacement in Problem 2. Does the volume change?
(c) Calculate the dilatation for the displacement u_x=Cx, u_y=Cy, u_z=Cz, where C is a constant. Does the volume change?
(d) Show that the dilatation is equal to the trace of the strain tensor (the trace of a matrix is the sum of the diagonal components) and also is equal to the divergence of the displacement (the divergence is defined in Chapter 4).

Problem 5 Consider the displacement u_x=Dx, u_y=-Dy, u_z=0, where D is a constant.
(a) Sketch a plot of the displacement distribution by drawing the displacement vectors over a 5 x 5 grid centered at the origin.
(b) Sketch how a small square in the x-y plane centered at the origin is deformed.
(c) Calculate the strain tensor (defined in Problem 3) for this displacement.
(d) Calculate the dilatation (defined in Problem 4) for this displacement.

Problem 6 Repeat the analysis of Problem 5 for the displacement u_x=Fy, u_y=Fx, u_z=0, where F is a constant.

Problem 7 Repeat the analysis of Problem 5 for the displacement u_x=Hy, u_y=-Hx, u_z=0, where H is a constant. This is a special case of rigid body motion. Interpret this displacement physically.

Problem 8 Like the strain, the stress can be written as a 3 x 3 symmetric tensor. For an isotropic material, the relationship between the components of the stress tensor, s_ij, and the strain tensor, ε_ij, is s_ij=λδ_ij(ε_xx+ε_yy+ε_zz)+2με_ij, where λ and μ are the Lame parameters, and δ_ij is the Kronecker delta (1 if i=j, and 0 otherwise).
(a) Show that for the case in Problem 1, this relationship reduces to Eq. 1.25 where s_n=s_xx and ε_n=ε_xx. Express the Young’s modulus E in terms of the Lame parameters.
(b) Show that for the case in Problem 2, this relationship reduces to Eq. 1.28 where s_s=s_xy and ε_s=2ε_xy. Express the shear modulus G in terms of the Lame parameters.
(c) Show that for the case in Problem 4c, this relationship reduces to Eq. 1.32 where the diagonal components of the stress tensor are given in terms of the pressure p as s_xx=s_yy=s_zz=-p. Express the compressibility κ in terms of the Lame parameters.

Problem 9 Figure 1.25 shows that pressure p will exert a net force on an element of fluid only if p is not uniform. Similarly, a stress will exert a net force on an element of tissue only if the stress is not uniform. The equations of mechanical equilibrium (zero net force) are ∂s_ix/∂x+∂s_iy/∂y+∂s_iz/∂z=0, where i is either x, y, or z.
(a) Substitute the relationship from Problem 8 into these equations, and derive three equations of mechanical equilibrium written in terms of the strain tensor.
(b) Substitute the relationships between the components of the strain tensor and the displacement found in Problem 3 and derive the equations of mechanical equilibrium in terms of the displacement.

Problem 10 Figure 1.20, showing a rod subject to a force along its length, is a simplification. Actually, the cross-sectional area of the rod shrinks as the rod lengthens. A better representation of the displacement than that given in Problem 1 would be u_x=Ax, u_y=-Aνy, and u_z=-Aνz, where A is a constant and ν is the Poisson’s ratio.
(a) Use the results of Problem 4 to calculate the dilatation.
(b) What value of Poisson’s ratio corresponds to an incompressible material (zero dilatation)?
(c) For an isotropic material, -1 « ν « 0.5. How would a material with negative ν behave?
Elliott et al. (2002) measured Poisson’s ratio for articular (joint) cartilage under tension and found 1 « ν « 2. This large value is possible because cartilage is anisotropic: its properties depend on direction.

Problem 11 Many biological tissues are composed mainly of water and are therefore nearly incompressible. To analyze such a tissue, start with the stress-strain relationship in Problem 8. (Assume u_z=0 and all derivatives in the z direction are zero; the case of “plane strain”.)
(a) For an incompressible tissue, ε_xx+ε_yy goes to zero and λ goes to infinity such that λ(ε_xx+ε_yy) is finite. Set it equal to –p, where p is the pressure.
(b) The displacement can be found from a stream function φ(x,y), where u_x=∂φ/∂y and u_y=-∂φ/∂x. Show that these definitions ensure that the dilatation is zero. Express the strain tensor in terms of φ.
(c) Write the components of the stress tensor (s_xx, s_yy, s_xy) in terms of p and φ.
(d) Use the analysis of Problem 9 to derive the equations of mechanical equilibrium in terms of p and φ.
(e) Manipulate these equations to find two new equations, one for p only and one for φ only. (Hint: try taking derivatives of the equations).

Problem 12 Start with the stress-strain relationship in Problem 8 and modify it to describe a two-dimensional sheet of cardiac muscle (Ohayon and Chadwick 1988). (Assume u_z=0 and all derivatives in the z direction are zero; the case of “plane strain”.)
(a) Cardiac tissue is nearly incompressible. For an incompressible tissue, ε_xx+ε_yy goes to zero and λ goes to infinity such that λ(ε_xx+ε_yy) is finite. Set it equal to –p, where p is the pressure.
(b) Cardiac muscle can develop an active tension T along the myocardial fibers caused by the interaction of actin and myosin molecules. Assume the fibers lie along the x direction, and add the term T to the expression for s_xx.
(c) The extracellular space consists of collagen fibers that can exert a shear force. Assume the collagen is isotropic, and interpret μ in Problem 8 as the collagen’s shear modulus.
(d) Derive expressions for s_xx, s_yy, and s_xy in terms of p, μ, T, and the strain tensor.
(e) Assume a solution u_x=-Ax, u_y=Ay, and p=P, where A and P are constants. If the tissue is free at its edges, then it must have zero stress throughout. Use this condition to derive expressions for A and P in terms of T and μ.
(f) Let T=3 x 10⁴ Pa and μ=10⁴ Pa (typical for cardiac tissue). Calculate values for A and P. Are the strains small? Sketch qualitatively the displacement distribution.

Ohayon, J. and R. S. Chadwick (1988) “Effects of Collagen Microstructure on the Mechanics of the Left Ventricle,” Biophys. J., Volume 54, Pages 1077–1088.